Best Known (44, 44+12, s)-Nets in Base 5
(44, 44+12, 2605)-Net over F5 — Constructive and digital
Digital (44, 56, 2605)-net over F5, using
- 51 times duplication [i] based on digital (43, 55, 2605)-net over F5, using
- net defined by OOA [i] based on linear OOA(555, 2605, F5, 12, 12) (dual of [(2605, 12), 31205, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(555, 15630, F5, 12) (dual of [15630, 15575, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(555, 15631, F5, 12) (dual of [15631, 15576, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(555, 15625, F5, 12) (dual of [15625, 15570, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(50, 6, F5, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(555, 15631, F5, 12) (dual of [15631, 15576, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(555, 15630, F5, 12) (dual of [15630, 15575, 13]-code), using
- net defined by OOA [i] based on linear OOA(555, 2605, F5, 12, 12) (dual of [(2605, 12), 31205, 13]-NRT-code), using
(44, 44+12, 7905)-Net over F5 — Digital
Digital (44, 56, 7905)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(556, 7905, F5, 12) (dual of [7905, 7849, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(556, 15632, F5, 12) (dual of [15632, 15576, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(555, 15631, F5, 12) (dual of [15631, 15576, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(555, 15625, F5, 12) (dual of [15625, 15570, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(50, 6, F5, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(555, 15631, F5, 12) (dual of [15631, 15576, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(556, 15632, F5, 12) (dual of [15632, 15576, 13]-code), using
(44, 44+12, 2499662)-Net in Base 5 — Upper bound on s
There is no (44, 56, 2499663)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1387 779258 445059 084284 969492 280846 884329 > 556 [i]